Thermodynamic quantum crtical behavior in the anisotropic Kondo necklace model

0.41 MB·English

by  D. Reyes

Thermodynamic quantum critical behavior of the
anisotropic Kondo necklace model
8
0
0
2 D. Reyes1, M. A. Continentino2, Han-Ting Wang3

n 1Centro Brasileiro de Pesquisas F´ısicas - Rua Dr. Xavier Sigaud, 150-Urca,
a
J 22290-180,RJ-Brazil
7 2Instituto de F´ısica, Universidade Federal Fluminense,
1 Campus da Praia Vermelha,

Nitero´i, RJ, 24.210-340,Brazil
]
l 3Beijing National Laboratory of Condensed Matter Physics and Institute of Physics,
e
- Chinese Academy of Sciences,
r
t Beijing 100080,People’s Republic of China
s
t. E-mail: [email protected]
a
m
Abstract.
-
d TheIsing-likeanisotropyparameterδintheKondonecklacemodelisanalyzedusing
n
the bond-operator method at zero and finite temperatures for arbitraryd dimensions.
o
c AdecouplingschemeonthedoubletimeGreen’sfunctionsisusedtofindthedispersion
[ relationfortheexcitationsofthesystem. Atzerotemperatureandintheparamagnetic

1 side of the phase diagram, we determine the spin gap exponent νz 0.5 in three
≈
v dimensions and anisotropy between 0 δ 1, a result consistent with the dynamic
2 ≤ ≤
exponent z = 1 for the Gaussian character of the bond-operator treatment. At low
9
7 but finite temperatures, in the antiferromagnetic phase, the line of Neel transitions is
2 calculated for δ 1 and δ 1. For d > 2 it is only re-normalized by the anisotropy
. ≪ ≈
1 parameter and varies with the distance to the quantum critical point QCP g as,
| |
0 T g ψ where the shift exponent ψ = 1/(d 1). Nevertheless, in two dimensions,
N
8 ∝ | | −
long rangemagnetic orderoccurs only at T =0 for any δ. Inthe paramagneticphase,
0
: we find a power law temperature dependence on the specific heat at the quantum
v
liquid trajectory J/t= (J/t) , T 0. It behaves as C Td for δ 1 and δ 1, in
i c → V ∝ ≤ ≈
X concordance with the scaling theory for z =1.
r
a
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 2
1. Introduction
Quantum phase transitions (QPT) from an antiferromagnetic AF ordered state to
a nonmagnetic Fermi liquid (NFL) in heavy fermion (HF) systems have received
considerable attention from both theoretical[1] and experimental points of view[2].
In contrast to classical phase transitions (CPT), driven by temperature, QPT can
be driven by tuning an independent-temperature control parameter (magnetic field,
external pressure, or doping). The physics of HF is mainly due to the competition
of two main effects: the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between
the magnetic ions which favors long-range magnetic order and the Kondo effect which
tends to screen the local moments and produce a nonmagnetic ground state. These
effects are contained in the Kondo lattice model (KLM) Hamiltonian in which, only spin
degreesoffreedomareconsidered. Hereweinvestigatedasimplifiedversion,theso-called
Kondo necklace model[3] (KNM) which for all purposes can be considered yield results
similar to the original model. While the ground state properties of this model has been
investigated rather extensively, by a variety of methods[4, 5, 6, 7,8, 9, 10, 11, 12, 13,14],
thermodynamicandfinitetemperaturecriticalproperties, closetoamagneticinstability,
remain an open issue. That was warned for us, and it was our first motivation for
studying the quantum critical properties of this model, as a function of the distance
to the quantum critical point g at zero and low temperatures[1, 15]. We extend now
| |
this treatment to finite inter-site anisotropy δ such that, 0 δ 1 since the δ = 1
≤ ≤
case is appropriate to describe compounds where the ordered magnetic phase has a
strong Ising component. However, the main reason for considering anisotropy δ in
the KNM is to try to describe its effects in the neighborhood of a magnetic quantum
critical point (QCP) in HF systems, rather than a symmetry problem[12]. This is a goal
in HF systems, and already several theories were formulated to explain their unusual
properties[16, 17, 18]. Besides, in an early work we were succeeded in finding that the
Neel line exists since turning on a geometric anisotropy[19], stressing that anisotropy
is an inherent ingredient in real HF systems. Henceforth, the model will be called
anisotropic Kondo necklace model (AKNM). This model was already investigated using
the real space renormalization group machinary[20] but just in one dimension and zero
temperature. We use the bond-operator approach introduced by Sachdev and Bhatt[21]
which was employed previously to both, KLM[22] and KNM[11] models but always at
(T,δ) = (0,0). We find that this method yields a shift exponent that characterizes
the shape of the critical line in the neighborhood of the QCP, as well as, the power
law temperature dependence on the specific heat along the so-called quantum critical
trajectory J/t = (J/t) , T 0. We consider the following AKNM:
c
→
H = t (τxτx +(1 δ)τyτy)+J S .τ , (1)
i j − i j i i
<i,j> i
X X
where τ and S are independent sets of spin-1/2 Pauli operators, representing the
i i
conduction electron spin and localized spin operators, respectively. The sum i,j
h i
denotes summation over the nearest-neighbor sites. The first term mimics electron
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 3
propagation which strength t and the second term is the magnetic interaction between
conduction electrons and localized spins S via the Kondo exchange coupling J (J > 0).
i
The Ising-like anisotropy parameter δ varies from the full anisotropic case δ = 1 to the
well established case δ = 0. Considering the bond-operator representation for two spins
S = 1/2, τ (S )α = 1(s†t + t† s iǫ t† t ) (α = x,y,z)[21], the Hamiltonian
i i ∓2 i i,α i,α i ± αβγ i,β i,γ
above, at half-filling, i.e., with one conduction electron per site, can be simplified and
the resulting effective Hamiltonian H with only quadratic operators is sufficient to
mf
describe exactly the quantum phase transition from the disordered Kondo spin liquid
to the AF phase, as discussed below. Then, we have a mean-field Hamiltonian:
3
H = N Js2 +µs2 µ +ω t† t
mf −4 − 0 k,z k,z
(cid:18) (cid:19) k
X
+ Λ t† t +∆ t† t† +t t
k k,x k,x k k,x −k,x k,x −k,x
Xk h (cid:16) (cid:17)i
+ Λ′ t† t +∆′ t† t† +t t , (2)
k k,y k,y k k,y −k,y k,y −k,y
Xk h (cid:16) (cid:17)i
where Λ = ω + 2∆ , Λ′ = ω + 2∆′ , ∆ = 1ts2λ(k), ∆′ = 1ts2λ(k)(1 δ)
k 0 k k 0 k k 4 k 4 −
and λ(k) = d cosk . s is the singlet order parameter consistent with the strong
s=1 s
coupling limit J/t , where the model becomes trivial, since each S spin captures a
P
→ ∞
conduction electron spin to form a singlet, and where the ground state corresponds to
a direct product of those singlets. The chemical potential µ was introduced to impose
the constraint condition of single occupancy, N is the number of lattice sites and Z is
the total number of the nearest neighbors on the hyper-cubic lattice. The wavevectors
k are taken in the first Brillouin zone and the lattice spacing was assumed to be unity.
This mean-field Hamiltonian can be solved using the Green’s functions to obtain the
thermal averages of the singlet and triplet correlation functions. These are given by,
(ω2 ω′2)(ω +Λ )
t ;t† = − k k ,
≪ k,x k,x ≫ 2πξ
(ω2 ω2)(ω +Λ′)
t ;t† = − k k ,
≪ k,y k,y ≫ 2πξ
1
t ;t† = , (3)
≪ k,z k,z ≫ 2π(ω ω )
0
−
where ξ = (ω2 ω2)(ω2 ω′2). The poles of the Green’s functions determine the
− k − k
excitation energies of the system as ω = J +µ , which is the dispersionless spectrum
0 4
of the longitudinal spin triplet states, ω(cid:16) = (cid:17) Λ2 (2∆ )2 that correspond to the
k ± k − k
excitation spectrum of the x-transverse spin triqplet states and ω′ = Λ′2 (2∆′)2
k ± k − k
that correspond to the y-transverse one. q
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 4
2. Paramagnetic State
From these modes above and their bosonic character an expression for the paramagnetic
internal energy at finite temperatures can be easily obtained[1, 15, 19],
U = ε + (ω n(ω )+ω n(ω )+ω′n(ω′)) (4)
0 0 0 k k k k
k
X
where ε = N 3Js2 +µs2 µ + (ω +ω′ Λ Λ′ )/2isthe paramagneticground
0 −4 − k k k− k− k
state energy, n(cid:16)(ω) = 1 coth βω(cid:17) 1Pthe Bose factor, β = 1/k T, k the Boltzman’s
2 2 − B B
constant and T the tem(cid:16)perature. A(cid:17)fter some straightforward algebra[1, 19] using Eq.
(4), the paramagnetic free energy renders
1 1 N
F = ε ln[1+n(ω )] ln[1+n(ω′)] ln[1+n(ω )]. (5)
0 − β k − β k − β 0
k k
X X
For obtaining s2 and µ we minimize the free energy by the saddle-point equations
1 Λ βω Λ′ βω′
2(2 s2) = k coth k + k coth k +f(ω ),
− 2N k ωk 2 ωk′ 2 ! 0
X
2J 3 µ 1 ω βω ω βω′
= 0λ(k)coth k + 0λ(k)(1 δ)coth k ,
t (cid:18)4 − J(cid:19) 2N k ωk 2 ωk′ − 2 !
X
(6)
where f(ω ) = N coth βω0 1 .
0 2 2 −
(cid:16) (cid:17)
2.1. Numerical results at T = 0
We first study the case T = 0 e.i., without thermal excitations. At zero temperature
the self-consistent equations given by Eqs. (6) can be simplified as,
4(2 s2) = I (y)+I (y)+I (y)+I (y)
1 2 3 4
−
4Jy 3 µ
= I (y) I (y)+I (y) I (y), (7)
2 1 4 3
t 4 − J − −
(cid:18) (cid:19)
with
1 π ddk 1 π ddk
I (y) = , I (y) =
1 3
πd Z0 1+yλ(k) πd Z0 1+y(1 δ)λ(k)
−
1 π q 1 qπ
I (y) = ddk 1+yλ(k), I (y) = ddk 1+y(1 δ)λ(k), (8)
2 πd 4 πd −
Z0 q Z0 q
where we have introduced a dimensionless parameter y = ts2/ω . An equation about y
0
can then be obtained:
2t
y = (1 [I (y)+I (y)]/4). (9)
1 3
J −
We will now obtain the numerical solutions to the zero temperature self-consistent
equations (7) using Eq. (9). In this case (paramagnetic phase), we have the z-polarized
branch of excitations has a dispersionless value ω (k) = ω and the other two branches
z 0
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 5
show a dispersion which has a minimum at the AF reciprocal vector Q = (π,π,π) in
three dimensions (3d). The minimum value of the excitations defines
∆x = ω 1 yd, ∆y = ω 1 yd(1 δ). (10)
0 0
− − −
q q
The spin gap energy ∆x and ∆y define the energy scale for the Kondo singlet phase, for
0 δ 1 and δ < 0 respectively. For δ = 0, ∆x and ∆y are identical and we obtain the
≤ ≤
original spin gap in the KNM[11]. Although we are interested in the case 0 < δ 1, we
≤
consider δ < 0 due to theoretical reasons. This case only will be consider at T = 0 and
will not be sketched in this report. The analysis of the spin gap is important because
the vanishing of gap and the appearance of soft modes define the transition from the
disordered Kondo spin liquid to the AF phase at the QCP (J/t = (J/t) ,T = 0). At
c
this point, it is suitable to clarify that in figures (1), (2) and (3), we sketched the spin
gap energy like ∆/J versus t/J by following the δ = 0 case[11], despite we consider
throughout this paper the control parameter as J/t. That will not yield any physical
difference since it only will change the onset of the curves from the left to the right.

1.0
0
0.01
0.04
0.1
0.4
J0.5 1
/
0.0
0 1 2 3 4 5
t/J
Figure 1. (Color online) The spin gap ∆/J vs the control parameter t/J is sketched
for different values of δ in one dimension and T =0. It shows that spin gap is always
nonzero for 0 δ 1.
≤ ≤
In the one dimensional (1d) case, the energy gap falls linearly for small values of
t/J and deviates considerably from the linear behavior as t/J gets larger, as it is relates
in Fig (1). Thereby, it is always nonzero for any δ, supporting its disordered phase, own
of 1d Kondo lattices[11, 23].
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 6

1.0
0
0.02
0.1
1
J
/
0.5
0.1
0.08J
/
0.06
0.02 0.025 0.03
I(t/J)C-t/JI
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t/J
Figure2. (Coloronline)Sketchatzerotemperatureofthespingapvsthestrengtht/J
intwodimensions. Theinsetshowsthelog-logplotofthespingapversus (t/J) t/J
c
| − |
for 0 δ 1. It shows that ∆/J vanishes close to (t/J) with a exponent νz 1.
c
≤ ≤ ≈
The anisotropy dependence on the spin gap in two dimensions (2d) is sketched
in Fig. (2). For 0 < δ 1 the effect of anisotropy is still weak and it changes the
≤
QCP slightly, until δ = 0, where both soft modes, ∆x and ∆y, contribute and the QCP
undergoes a slight jump. Then, the qualitative behavior is the same for this range and
the gap exponent is approximately νz 1. It is plotted in the inset of Fig. (2). On the
≃
other hand, for δ < 0, the QCP is reduced and the Kondo spin liquid phase is limited
to a narrower region. It is not shown in Fig. (2).
In three dimensions, the effect of anisotropy on the spin gap is similar as in the
2d case. The spin gap follows a exponent νz 0.5 for 0 δ 1 and changes its
≃ ≤ ≤
universality for δ < 0, where the spin gap vanishes close to the QCP more faster. As in
the 2d case, there is a jump for δ = 0 which is the result of all soft modes coincide.
We conclude that, for all anisotropy between 0 δ 1, there exists a critical
≤ ≤
value (t/J) , where the spin gap vanishes as ∆/J (t/J) t/J νz, and a QPT to
c c
∝ | − |
the ordered magnetic phase occurs in 2d and 3d whereas no transition happens in 1d.
This is similar to the results in Ref. [11] for δ = 0 and it gives us a kind of universality
similar as in the isotropic Kondo lattices[1, 11, 22]. From relation between the spin gap
and the distance to the QCP, sketched in the onset of Fig. (3), it is shown that when
t/J increases from its strong coupling limit, the triplet spin gap at the wave vector
Q = (π,π,π) decreases and vanishes at t/J = (t/J) . Since ∆/J (t/J) t/J 0.5,
c c
∝ | − |
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 7

1.0
0
0.1
0.4
1

0.2
J
/ 0.5
0.18
0J.16
/
0.14
0.12
0.01 0.015 0.02
I(t/J)C-t/JI
0.0
0.0 0.1 0.2 0.3 0.4
t/J
Figure 3. (Color online) Anisotropy dependence on the spin gap vs the strength t/J
in three dimensions at T = 0 and 0 δ 1. The scaling of gap close to the QCP is
≤ ≤
shown in the inset of the figure. It shows the log-log plot of ∆/J vs (t/J) t/J for
c
| − |
δ = 0,0.1,0.4,1, and scales close to the QCP like ∆/J (t/J) t/J νz with spin
c
∼ | − |
gap exponent νz 0.5.
≈
close to the QPT, we can immediately identify the spin gap exponent νz 0.5 at the
≈
QCP of the Kondo lattice, confirming our early theoretical results[1]. Finally, for δ < 0
exists also a QPT in d = 2,3 but no phase transition appears in 1d.
2.2. Analytical results at the quantum critical trajectory
Since quantum phase transitions are generally associated with soft modes at the
QCP, where the gap for excitation vanishes, then physical quantities have power law
temperature dependencies determined by the quantum critical exponents[24]; one of
them is the specific heat C , that we will calculate here. This strategy has been
V
intensively explored in the study of heavy fermion materials, in the so-called quantum
critical trajectory J/t = (J/t) , T 0, fixing the pressure (in our case the control
c
→
parameter J/t) at its critical value for the disappearance of magnetic order[25]. Then,
wecalculate analytically, theanisotropydependence onthespecific heat atJ/t = (J/t) ,
c
T 0 forboth cases, δ 1 andδ 1. Allthe calculations will be done considering two
→ ≪ ≈
essential approximations: (i) The system is at the quantum critical point J/t = (J/t) ,
c
and temperatures T 0. (ii) The temperatures region where will be found the specific
→
heat will be lower than the Kondo temperature (T ). We will begin writing k = Q+q
K
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 8
and expanding for small q: λ(q) = d + q2/2 + O(q4), this yields the spectrum of
−
transverse spin triplet excitations as,
ω ω 1+yλ(q) = ∆2 +Dq2,
q 0
≈
q q
ω′ ω 1+yλ(q)(1 δ) = ∆2 +D(1 δ)q2 +ω2δ, (11)
q ≈ 0 − − 0
q q
where ∆ = ∆x is the spin gap energy given by Eq. (10) since 0 δ 1, D = ω2/2d
≤ ≤ 0
the spin-wave stiffness at T = 0, and ω is the z-polarized dispersionless branch of
0
excitations. Considering ∆ = 0, at the QCP[24] in the spectrum excitations Eq. (11),
and using C = T∂2F/∂T2 in Eq. (5) we get
V
−
S π βω βω′
C = d dqqd−1(ω2+ω′2)(sinh−2 q +sinh−2 q), (12)
V 4k T2πd q q 2 2
B Z0
where S is the solid angle. Equation (12) yields the expression for the anisotropic
d
dependence on specific heat at the quantum critical trajectory, as an contribution of
bosons t and t .
x y
Case 0 δ 1—Having shown the relationship between the specific heat C and
V
≤ ≪
δ we now discuss the case δ 1. Making change of variables in Eq. (12) we obtain,
≪
S k Zd/2 k T d δ
d B B
C (δ 1) = [Υ (d)+ (Υ (d) 2Υ (d))], (13)
V ≪ πd ω0 ! 1 4 2 − 1
where Υ (d) = ∞dxxd+1sinh−2(x/2), Υ (d) = ∞dxxd+2coth(x/2)sinh−2(x/2) and
1 0 2 0
x = βω q/√Z. In two dimensions we found Υ (2) = 24ζ(3) and Υ (2) = 96ζ(3),
0 R R1 2
where ζ is the Riemann zeta-function. In three dimensions Υ (3) = 16π4/15 and
1
Υ (3) = 16π4/3. For δ = 0, the spectrum excitations given by Eq. (11) coincide and we
2
recover the exact value as obtained in an previous work for the isotropic KNM[1].
Case δ 1—Here, it is sufficient to consider, ξ = 1 δ 1, where ξ is a
≈ − ≪
dimensionless parameter that controls the Ising-like anisotropy in this case. Thereby,
working in analogy with the preceding case, we obtain
S k Zd/2 k T d
d B B
C (δ 1) = Υ (d)(2 δ), (14)
V ≈ 4πd ω0 ! 1 −
where we have already replaced the ξ expression. The results above show that the
specific heat of the AKNM for δ 1 and δ 1 is only re-normalized by the anisotropy,
≪ ≈
concluding that C Td at the quantum critical trajectory for δ 1 and δ 1.
V
∝ ≪ ≈
Notice that this is consistent with the general scaling result C Td/z with the
V
∝
dynamic exponent taking the value z = 1[24]. Since z = 1, in three dimensions
d = d + z = d = 4 where d is the upper critical dimension for the magnetic
eff c c
transition [24]. Consequently, the present approach yields the correct description of the
quantum critical point of the Kondo lattices for d 3.
≥
3. Antiferromagnetic Phase
The mean-field approach can be extended to the AF phase assuming the condensation
in the x component of the spin triplet like: t = √Nt¯δ +η , where t¯is its mean
k,x k,Q k,x
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 9
value in the ground state and η represents the fluctuations. Making the same steps
k,x
as before, the internal energy renders
U′ = ε′ + (ω n(ω )+ω n(ω )+ω′n(ω′)), (15)
0 0 0 k k k k
k
X
where ε′ = N 3Js2 +µs2 µ+ J +µ 1tZs2 t2 + (ω + ω′ Λ Λ′ )/2 is
0 −4 − 4 − 2 k k k − k − k
the AF groundhstate. The free ener(cid:16)gy is now (cid:17) i
P
1 1 N
F′ = ε′ ln[1+n(ω )] ln[1+n(ω′)] ln[1+n(ω )].(16)
0 − β k − β k − β 0
k k
X X
Minimizing the free energy Eq. (16), using (∂F′/∂µ,∂F′/∂s,∂F′/∂t¯) = (0,0,0), we can
easily get the following saddle-point equations,
J f(ω )
s2 = 1+ 0
Zt − 2
1 2λ(k) 2λ(k)(1 δ)
1+ (1+2n(ω ))+ 1+ − (1+2n(ω′)) ,
− 4N s Z k s Z k 
k
X
 
J f(ω ) 1 (1+2n(ω )) (1+2n(ω′))
t2 = 1 0 k + k ,
− Zt − 2 − 4N k  1+ 2λ(k) 1+ 2λ(k)(1−δ)
X Z Z
1  q q 
µ = Zts2 J/4, (17)
2 −
with the excitations spectrum of the x-transverse and y-transverse spin triplet states
given now by, ω = 1Zts2 1+2λ(k)/Z and ω′ = 1Zts2 1+2λ(k)(1 δ)/Z,
k 2 k 2 −
respectively. Generally the eqquations for s and t in Eq. (17) qshould be solved and
for δ = 0 the results of Ref. ([1]) are recovered. Here, in the magnetic ordered state, the
condensation oftriplets(singlets) followsfromtheRKKYinteraction(Kondoeffect). At
finitetemperaturesthecondensationofsingletsoccursatatemperaturescalewhich, toa
first approximation, tracks theexchange J while theenergy scale below which the triplet
excitations condense isgiven bythecriticalNeeltemperature (T )which iscalculated in
N
the next section. Thus, the fact that at the mean-field level, both s and t do not vanish
may be interpreted as the coexistence of Kondo screening and antiferromagnetism in
the ordered phase[1, 11, 22] for all values of the ratio J/t < (J/t) .
c
4. Critical line in the AKNM
Following the discussion above, the critical line giving the finite temperature instability
of the AF phase for J/t < (J/t) is obtained making t = 0. Hence, from Eq. (17) we
c
can obtain the boundary of the AF state as,
g 1 n(ω ) n(ω′) f(ω )
| | = k + k + 0 , (18)
Z 2N k  1+ 2λ(k) 1+ 2λ(k)(1−δ) 2
X Z Z
q q 
where g = (J/t) (J/t) measures the distance to the QCP. The latter is
c
| − |
given by, (J/t) = Z[1 1 ( 1 + 1 )], which separates an
c − 4N k √1+2λ(k)/Z √1+2λ(k)(1−δ)/Z
P
Thermodynamic quantum critical behavior of the anisotropic Kondo necklace model 10
antiferromagneticlongrangeorderedphasefromagappedspinliquid phase. Performing
the same analysis as in sub-section (2.2), expanding the spectrum excitations close to
Q = (π,π,π), Eq. (18) becomes
g S ω π 1 βω 1 βω′
| | = d 0 dqqd−1 coth q 1 + coth q 1 , (19)
Z 4πd Z0 ωq 2 − ! ωq′ 2 − !!
where we have considered that for temperatures k T ω , f(ω ) goes to zero faster
B 0 0
≪
than the first term of Eq. (18). This equation above allow us to obtain the critical line
in the AKNM as a function of the anisotropy parameter δ.
4.1. Case 0 δ 1
≤ ≪
We now demonstrate analytically the appearance of a finite Neel line temperature when
a small degree of anisotropy δ in y-component spin is turned on. Then, solving Eq. (19)
for 0 δ 1, we get
≤ ≪
g S Zd/2 k T d−1 δ
d B
| | = Φ (d)+ (Φ (d)+2Φ (d)) , (20)
Z δ≪1 2πd ω0 ! " 1 8 2 1 #
where Φ (d) = ∞dxxd−2 coth x 1 and Φ (d) = ∞dxxd+1sinh−2(x/2). We
1 0 2 − 2 0
notice that the inRtegrals Φ1((cid:16)d) and Φ2((cid:17)d) diverge for d <R 3 showing that there is no
critical line in two dimensions at finite temperatures[1, 15] for any anisotropy δ 1,
≪
in agreement with the Mermin-Wagner theorem[26]. Nevertheless, for d 3, the
≥
integrals are finite and the equation for the critical line shows, (T ) g φ, with
N δ≪1
∝ | |
φ = 1/(d 1). If we write the equation for the critical line, f(g,T) = 0, in the form,
−
(J/t) (T) (J/t) (0) + v T1/ψ = 0, with v related to the spin-wave interaction, we
c c 0 0
−
identify the shift exponent, ψ = z/(d+ z 2)[27], that comparing with φ gives us the
−
dynamic exponent z = 1, a Gaussian result, since the critical line only exists for d > 2.
The temperature dependence of the function f arising from the spin-wave interactions
can modify the temperature dependence of the physical properties, as the specific heat,
at J/t = (J/t) . However, in the limit T 0 we can easily see that the purely Gaussian
c
→
results for the specific heat calculated in section (2.2) is dominant, in concordance with
the mean-field treatment used here. For δ = 0, we obtain the well established result for
the critical line in the KNM[1], this due to the fact that the spectrum energy of the two
excitations coincide. For δ 1, following the same steps as before, it is straightforward
≈
to show that the dominant is also T (δ 1) g φ for d 3, and no critical line exists
N
≈ ∝ | | ≥
for d = 2.
In summary, we have obtained analytically the expression for the Neel line, below
which the triplet excitations condense, close to the QCP for 0 δ 1 and δ 1. We
≤ ≪ ≈
have shown that this line does not exist for d = 2 for any value of the anisotropy, as
we expected, whereas for d 3, the power dependence on g of the critical line in the
≥ | |
presence of the anisotropy is the same of the KNM original. Therefore, the criticality
close the QCP is governed by the same critical exponents of the isotropic δ = 0 case
that we have calculated before[1].